Editing Symmetric difference (section)

{{short description|Elements in exactly one of two sets}}
{{Infobox mathematical statement
| name = Symmetric difference
| image = Venn0110.svg
| caption = [[Venn diagram]] of <math>A  \Delta B</math>.  The symmetric difference is the [[Union (set theory)|union]] [[Complement (set theory)#Relative complement|without]] the [[Intersection (set theory)|intersection]]:
{{nowrap|[[File:Venn0111.svg|30px]] <math>~\setminus~</math> [[File:Venn0001.svg|30px]] <math>~=~</math> [[File:Venn0110.svg|30px]]}}
| type = [[Set (mathematics)#Basic operations|Set operation]]
| field = [[Set (mathematics)|Set theory]]
| statement = The symmetric difference is the set of elements that are in either set, but not in the intersection.
| symbolic statement = <math>A\, \Delta\,B = \left(A \setminus B\right) \cup \left(B \setminus A\right)</math>
}}

In [[mathematics]], the '''symmetric difference''' of two [[Set (mathematics)|sets]], also known as the '''disjunctive union''' and '''set sum''', is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets <math>\{1,2,3\}</math> and <math>\{3,4\}</math> is <math>\{1,2,4\}</math>.

The symmetric difference of the sets ''A'' and ''B'' is commonly denoted by <math>A \operatorname\Delta B</math> (alternatively, <math>A \operatorname\vartriangle B</math>), <math>A \oplus B</math>, or <math>A \ominus B</math>.
It can be viewed as a form of [[Modular arithmetic|addition modulo 2]].

The [[power set]] of any set becomes an [[abelian group]] under the operation of symmetric difference, with the [[empty set]] as the [[neutral element]] of the group and every element in this group being its own [[inverse element|inverse]]. The power set of any set becomes a [[Boolean ring]], with symmetric difference as the addition of the ring and [[intersection (set theory)|intersection]] as the multiplication of the ring.